OFFSET
1,2
COMMENTS
The sums-complement of a sequence s(1), s(2), ... of positive integers is introduced here as the set of numbers c(1), c(2), ... such that no c(n) is a sum s(j)+s(j+1)+...+s(k) for any j and k satisfying 1 <= j <= k. If this set is not empty, the term "sums-complement" also applies to the (possibly finite) sequence of numbers c(n) arranged in increasing order. In particular, the difference sequence D(r) of a Beatty sequence B(r) of an irrational number r > 2 has an infinite sums-complement, abbreviated as SC(r) in the following table:
r B(r) D(r) SC(r)
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From Jeffrey Shallit, Aug 15 2023: (Start)
Simpler description: this sequence represents those positive integers that CANNOT be expressed as a difference of two elements of A022839.
There is a 20-state Fibonacci automaton for the terms of this sequence (see a276871.pdf). It takes as input the Zeckendorf representation of n and accepts iff n is a member of A276871. (End)
LINKS
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See p. 16.
Jeffrey Shallit, Fibonacci automaton for A276871
EXAMPLE
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Sep 24 2016
STATUS
approved